The Art Institute is Chicago's world-class Art Museum. It has particular strengths in 19th century art, including a really good Impressionist collection, but it has a large enough collection overall to do justice to just about every era. The Art Institute is free on Tuesdays, and the rest of the time, it's 'suggested donation' based.
I highly recommend starting on the 2nd floor of the main building - a signature item in their 19th century collection is Seurat's La Grande Jatte which you've probably seen somewhere before. Surrounding that is the usual collection of Monets, Manets, Renoirs, et al.
Park underground at Millenium Park or by the meters nearby.
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